1.5 Moments, Skewness, and Kurtosis

1.5.1 Central Moments

Central Moments: \(\mu_r=\frac{\sum(x_i-\bar x)^r}{n}\)

  • \(\mu_1=\frac{\sum(x_i-\bar x)}{n}=\frac{\sum x_i}{n}-\frac{n \bar x}{n}=\frac{n \bar x}{n}-\frac{n \bar x}{n}=0\)

  • \(\mu_2=\frac{\sum(x_i-\bar x)^2}{n}=\sigma^2\)

  • \(\mu_3=\frac{\sum(x_i-\bar x)^3}{n}\)

  • \(\mu_4=\frac{\sum(x_i-\bar x)^4}{n}\)

  • For grouped data: \(\mu_r=\frac{\sum f_i(x_i-\bar x)^3}{n}\)

1.5.2 Raw Moments

\(\mu_r=\frac{\sum(x_i-a)^r}{n}\); a is arbitrary number